Interplay of valley, layer and band topology towards interacting quantum phases in moiré bilayer graphene

In Bernal-stacked bilayer graphene (BBG), the Landau levels give rise to an intimate connection between valley and layer degrees of freedom. Adding a moiré superlattice potential enriches the BBG physics with the formation of topological minibands — potentially leading to tunable exotic quantum transport. Here, we present magnetotransport measurements of a high-quality bilayer graphene–hexagonal boron nitride (hBN) heterostructure. The zero-degree alignment generates a strong moiré superlattice potential for the electrons in BBG and the resulting Landau fan diagram of longitudinal and Hall resistance displays a Hofstadter butterfly pattern with a high level of detail. We demonstrate that the intricate relationship between valley and layer degrees of freedom controls the topology of moiré-induced bands, significantly influencing the energetics of interacting quantum phases in the BBG superlattice. We further observe signatures of field-induced correlated insulators, helical edge states and clear quantizations of interaction-driven topological quantum phases, such as symmetry broken Chern insulators.

/ 0 = 1/ lines, the first order Brown-Zak oscillation occurs and the sign of   is reversed as the effective magnetic field felt by the BZ quasiparticle is reversed.In addition, horizontal lines with   close to zero are observed at the higher order BZ oscillation (/ 0 = 2/) lines (Black arrows in d,e).Supplementary Figure 3 | Displacement field dependent CNP states a, Reciprocal of Longitudinal conductance in units of  2 /ℎ as a function of applied perpendicular electric displacement field and magnetic field at  = 0 (CNP) and T = 30 mK.Labels (i) and (ii) indicate the two different insulating  = 0 phases; (i) canted antiferromagnet state and (ii) layer polarized insulator state known in pristine BBG. 1 There is a phase transition point between the two phases where the conductance increases.But, as the magnetic field increases, this phase transition point widens and a metallic region appears between (i) and (ii).b, A numerical simulation of gap closings (with the marker size inversely proportional to the gap size) near CNP for the same conditions is presented for comparison, showing a good agreement with the measurement.This gap closing is likely due to the overlapping of the broadened conduction and valence bands under the influence of the moiré potential.

Supplementary Note 3. Transport data axis normalization and superlattice size estimation.
The x-axis of the raw data in the Landau fan diagram is the gate voltage proportional to the carrier density, and the y-axis is the perpendicular magnetic field.Where e is the unit charge,  0 is the permittivity of vacuum,   and   are the capacitance per unit area between the sample and the top gate and bottom gate respectively.If we think of (  ,   ) as a single two-dimensional vector and call the norm of this vector c and the polar angle , we can define   , the voltage proportional to the carrier density, and   , the voltage proportional to the  field, as above.Then, if we know  and c, we can find the carrier density and  field from the voltage.First, to measure , a longitudinal resistance measurement was performed by sweeping the top gate voltage and bottom gate voltage in the magnetic field of 0.5T.The darker colored regions with lower resistance are the integer quantum Hall states with chiral edge modes, and the lighter regions in between are CNP (brightest line) and the metallic states with partially filled Landau level extended states.The direction perpendicular to these diagonal lines is the direction of increasing carrier density, and by measuring the angle between this direction and the x-axis (   axis),  defined above can be obtained, and  = 0.505rad was measured.From this, it can be seen that   = 0.5528  .By fixing   ,  field can be fixed and only the carrier density can be tuned. .So the slope of   () is proportional to the reciprocal of the carrier density, which is calculated and plotted for each   .Error bars are proportional to the slope error of   () at each   .From the linear fit, we can see that  = 0.6989  − 0.0074 (10 12  −2 ), and using the fact that the coefficient of   is /, we can see that  = 1.1198 × 10 −15   −2 .

Supplementary
The Hall measurement was performed in the configuration shown in Supplementary Figure 2a, and since it is not an ideal Hall bar geometry, it measures a mixed value of   and   , but at the Drude model level,   is independent of the magnetic field, so the slope can be used to determine the carrier density.From the measurement results, we determined the relationship between carrier density n and   .From Supplementary Figure 12, we can see that a band insulator appears at   = −3.42.This can occur when the moiré superlattice is filled with all four flavors of spin and valley.The superlattice parameter  can be calculated from the corresponding carrier density and the fact that the moiré superlattice is a triangular lattice.

Supplementary
From the magnitude of the superlattice parameter, we can see that our sample has BBG and hBN aligned at almost 0 deg.
Alternatively, the superlattice parameter can be obtained independently by observing the Brown-Zak oscillation.From Fig. 2 and Supplementary Figure 2, / 0 = 1/ line is equivalent to  = (24.7/) line, so we can estimate the superlattice unit cell area by  × 24.7 =  0 = ℎ/,  = √3 2  2 .The resulting  is 13.90 nm which is almost the same value obtained by the above method.

Supplementary Note 4. Numerical simulation of Hofstadter energy spectra and Wannier plots of the BBG/hBN superlattice with zero-degree alignment
After electronic energy eigenvalues are obtained by numerically diagonalizing at various perpendicular magnetic fields, we transform it to the Wannier plot as a function of density and magnetic field as in Supplementary Figure 13.It shows features that can be directly associated with our measured data.First of all, in the low magnetic fields, the LL band broadening due to the moiré potential is selectively strong for a certain valley (see also Supplementary Figure 15).As discussed in the main text, it is counterintuitive that the valley degrees of freedom determine the strength of the superlattice effect, not the layer degrees of freedom.
The multiple plots by varying the interlayer potential difference in Supplementary Figure 14 ━ where we define  field = (interlayer potential difference) / (interlayer distance) ━ shows the pronounced features that strongly depend on the  field.We reproduce the insulating gaps at the full fillings, the characteristic spectra of the ZLL and the valley degrees of freedom's dependence on the external vertical electric displacement field, etc. Importantly, the vertical insulating phases at / 0 = −2 and −4 measured in Fig. 4 of the main text are either absent or having energy gaps just similar to other incompressible Chern phases in the calculation.This thus supports our conclusion that the strong insulating phase at / 0 = 2 has an enhanced energy gap due to the electron correlation, and it is likely the consequence of the narrow bandwidth of the isolated valence band (see the calculated spectra in Supplementary Figure 13a-d), as discussed in the main text.
Longitudinal resistance   as a function of the carrier density n and D field at 8.9T(e), 11.75T (f), 12.4T (g).N denotes the orbital number of the Landau level of BBG and white numbers in a, d denote the Landau filling factor.Each colored horizontal line in e-f corresponds to the same colored lines in a-c.We fabricated a 1.37 deg aligned device similar to the device used in the main text.(top hBN: 67 nm, bottom hBN: 50 nm) The sample showed similar valley-selective moiré effects in each Landau level and D field tunability of Chern insulators.We observed integer quantum Hall, Chern insulator, and fractional quantum Hall (up to N=3 LL) states in the Hofstadter butterfly pattern of the new sample, but not symmetry broken Chern insulator and fractional Chern insulator states.For the device of the main text, interacting states appeared in the low (less than 1 × 10 12  −2 ) carrier density region, but the new device required about three times higher carrier densities to fill the same superlattice filling factor, so we could observe only the high density and low / 0 region of Hofstadter patterns in the magnetic field range of our system.We believe that this device will also show interaction driven states if the magnetic field is further increased to access the low carrier density region of the Hofstadter pattern.Supplementary Figure 19 | Magnetotransport measurement of other samples.a, Device with gate leakage.Only a small range of carrier density was tuned, and a transition to higher resistance in the ZLL metallic state was observed around B=11T. White number denotes the Landau filling factor of BBG.b, Device with bad contact resistance.Fabricated without contact graphite and using a flake with naturally connected trilayer graphene and BBG regions, the contact quality became bad with increasing magnetic field.
this state as a QVH state, in which the valleys have edge channels with opposite chirality, forming helical edge states.
We note that, since the spectra in Supplementary Figure 16 are limited to the spin degenerate case, a spin hall state or a complex spin-valley magnetic state are not considered and thus cannot be ruled out given the spin degeneracy is to be broken in the real system.Nonetheless, we think that the valley-dependent Chern number provides a crucial mechanism in the formation of the helical edge state.

Figure 4 |
−  sweep at various magnetic field.Longitudinal resistance versus carrier density and  field at a, B = 1 T, b, B = 3.8 T, c, B = 5.8 T, d, B = 7 T, e, B = 10 T, and f, B = 13.5 T. Numbers in each figure represent the Landau filling factor . a shows that at the low magnetic field, there is little effect from the moiré potential and a single particle Landau level spectrum of intrinsic BBGs with broken spin-valley degeneracy appears.Around D = 0, intra-Landau level crossings for each N were observed, and for large  field, inter-Landau level crossings between different Ns also occurred. 2b is the zoomed-out image of Fig. 3a.Two of four N = 3 LL feels the moiré potential while  ≤ 2 LLs are not yet affected by the moiré potential.In c,d, it is observed that the states with strongly feeling moiré potentials in the ZLL and N = 2 LL shift in opposite directions as D is varied.e,f shows that a large number of CIs undergo complex transitions, and the insulating gap at CNP closes in a certain  field range.Each color scale is truncated at the end value of its respective colorbar.Supplementary Figure 7 | Symmetry broken Chern insulator states in higher Landau level at zero displacement field.a,b A zoomed in image of longitudinal(Hall) resistance Landau fan diagram at D = 0 (See Supplementary Figure 1c and 2e).The color scale is truncated at 10 k in a. c, Wannier diagram of the same area as a, b.The upper SBCI has (t,s) = (-1,-3/2), (-3,-1/2), from left to right, and the lower SBCI has (-2,-2/3), (-3,-1/3), respectively.The upper SBCIs are half-filled with the band with t = -2, and the lower SBCIs are 1/3 and 2/3 filled with the band with t = -3, respectively.d-e, Line cuts along the upper (d), lower (e) dashed line in b.The colored arrows indicate the states corresponding to the same colored lines in c.Each of the incompressible states showed a well quantized Hall resistance except the CI (-1, -1) state in e which didn't reached ℎ/ 2 .This is probably due to a part of the   component (bright metallic part surrounding the gap) was not removed during the removal process, or the measurement speed didn't allow the value to saturate as the t value flipped from 2 to -1. f, Temperature dependence of longitudinal resistance along the dashed lines in a. g, Arrhenius plots of identified SBCIs.Supplementary Figure 8 | Nonlocal measurement of helical edge states at t=0 insulators.ab, Measurement configuration of longitudinal (  , a) and non-local (  , b) resistance.Scale bar, 15 μm.c-d, Longitudinal (c) and non-local (d) resistance, as a function of normalized carrier density and magnetic flux, measured at  = 30 mK with  = 91  −1 .e-f, Line cuts of longitudinal (e) and non-local (f) resistance along the magnetic field direction at / 0 = -1 (red), -2 (green), and -4 (blue).Dashed horizontal line in f represents ℎ/4 2 .

Figure 10 |
Capacitance ratio determination.a, Longitudinal resistance was measured by sweeping the top and bottom gate voltage at  = 0.5 T,  = 30 mK.Bright white diagonal line at the center is the CNP of BBG and each black line represents an integer quantum Hall state.By measuring the angle between   direction and grad() direction, we can get the ratio between the top and bottom capacitance. was 0.505 rad in our device.b, Schematic diagram of axis conversion.Blue and red lines perpendicular to   and   are the equipotential line of each variable. is the angle defined the same as in a.
And to measure c, low field Hall measurement was performed Supplementary Figure 11 | Capacitance size determination.a, Low field Hall resistance measurement.In the low carrier density region, quantum Hall is already visible below 1 T, but as the density increases,   shows a linear relationship as predicted by the Drude model.b, Measured   dependence of the carrier density.From the Drude model,   () = −

Figure 12 |
Determination of superlattice lattice parameter.a, Raw data of the Landau fan diagram.Longitudinal resistance was measured as   and B were swept.Superlattice band insulator was observed at   = −3.42.This corresponds to the carrier density  = 2.397 × 10 12  −2 .b, moiré superlattice pattern of 0 degree aligned BBG/hBN heterostructure.The lattice constant mismatch of about 1.8% generates a triangular superlattice structure with the lattice constant .
To convert this to superlattice filling factor and normalized flux density, top gate and bottom gate sweep and Hall measurement were performed at low field.The carrier density n and vertical electric displacement field  can be expressed as follows using the top gate voltage   and bottom gate voltage   .